Ah, this is good.
If you can shift eyes independently, then presumably you're shutting down only half your visual cortex at a time, so your consciousness is able to access a stationary view of the opposite eye moving.
Now we need you to try The_Radiation_Specialist's experiment, using this same trick eye movement. If his transient stationary view does have something to do with saccadic suppression, then we'd predict you wouldn't be able to see it.

Grant Hutchison